At some point, maybe thirty years ago, it became fashionable to emphasize functions and their multiple representations in secondary school math. This was in part driven by the newly available electronic graphing technology, and in part by the realization that important curricular ideas should be approached in more than one way. As is often the case in math education pendulum swings, some people took this too far, asserting that functions were the key to everything, and could for example be the unifying thread in Algebra 2 or even all of high school math. While I steered clear of this extreme interpretation, I welcomed this development, and multiple representations of functions became a key part of my toolbox.
The standard representations of functions are the algebraic formula, the Cartesian graph, and the input-output table (known by some as a “T-table”.) At some point, I came across the concept of function diagrams, and found that this fourth representation offers additional benefits at all levels from middle school to calculus. (See: Function Diagrams.)
The existence of multiple representations suggests the idea of challenging students to translate from one to another. For example, it is straightforward to go from formula to graph, either by using technology, or by plotting individual points and connecting them. However, it is pedagogically much more powerful to go in the other direction: given the graph, what is the formula that yielded it? I explored this in Make These Designs.
In today’s post, I’ll flesh out another rich translation challenge: going from table (or function diagram) to formula.
What’s My Rule?
Back in my K-5 days, this took the form of the widely-played “What’s My Rule?”. Students provide input numbers, and I respond with an output. They are to guess my function. Sometimes I built a story around a “function machine” which provided the output using a mechanical-sounding voice. Often, I would ask a student who guessed the formula to come up to the blackboard and “be” the machine.
Typically, students managed quite well as long as the function involved a single operation, but the game became challenging and interesting if there were two operations (e.g. “double the input, then subtract 3”).
A fun twist on this is when the function machine is a clock, as on page 12 of my Abstract Algebra packet, which works as an introduction to modular arithmetic.
Some Variations
“What’s My Rule?” also works well in middle school and high school.
One extension is “What’s in the Bag?” The game is described in detail on pp. 103-104 of Algebra Lab Gear: Algebra 1. The idea is to put a few Lab Gear blocks in a bag, and have students ask questions of the form “If x is 1 and y is 2, what’s in the bag?” The fact that we are now dealing with a function of two variables ups the ante considerably, and it is a good idea to start with warm-up challenges using only x and constant blocks.
Here are some more variations on “What’s My Rule?”, most of them involving function diagrams:
Constant Sums, Constant Products
In Algebra: Themes, Tools, Concepts Anita Wah and I offered some lessons in which we encouraged students to recognize situations where the x and the y have a constant sum or a constant product. In other words, recognize x + y = S and xy = P from table data. As it turns out, this yields a wealth of useful pedagogical and curricular payoffs, so many that I don’t have room to list them here.
See one application in this lab: Perspective.
Read more about all this: Constant Sums, Constant Products.
Functions from Patterns
I will end this post by introducing some powerful ideas I got from Paul Foerster’s very interesting Algebra 2 and Precalculus textbooks. He ends some chapters with what he calls “functions as mathematical models”, essentially word problems where one can apply some ideas about functions to go from a small number of data points to a formula connecting them. I have some differences with Foerster’s pedagogy, but here I want to focus on the mathematics.
He proposes diagnostic tests to distinguish different sorts of functions by analyzing their tables:
- Add-Add for a linear function: adding a constant to x adds a constant to y. (A consequence of the constant slope.)
- Multiply-Multiply for an nth power variation: multiplying x by a constant multiplies y by that constant raised to the nth power.
- Add-Multiply for an exponential function: adding a constant to x multiplies y by a constant.
- Multiply-Add for a log function: multiplying x by a constant adds a constant to y.
(Foerster does not use the above italicized names for the properties, but I found them useful as a shortcut when discussing them.)
There are many things I like about this. First of all, engaging with actual numbers helps students develop a feel for these functions to complement and explain what they see in graphs. Second, it provides an entry point to the broader issue of mathematical modeling: if you think a certain phenomenon can be modeled by a function, a good place to start is identifying which sort of function it might be. Third, at the precalculus level, proving these properties algebraically is a worthwhile challenge. Fourth, this understanding provides the foundation for the use of log and semilog graphs.
A legitimate objection to this technique is that you can only apply it to tables that have the right sort of data in them. For example, if you happen to have 2, 4, 6, 8, and 10 for the x values, as in the exercises Foerster supplies, then you can see what happens when adding 2 to the x, and in a couple of cases what happens when the x is doubled. Still, the idea is not that this is a universally applicable tool — it’s just a useful pedagogical device.
I mostly didn’t use Foerster’s book, but I did apply these ideas in my teaching. Though they are not comprehensive, see for example these worksheets on nth power variation, and their Teachers’ Guide.
The permanent home page for this general topic on my website is Recognizing Functions.